A circular wire of radius 2.5 cm is cut and bent to as to lie along the circumference of a hoop whose radius is 1.29 m. Find (in degrees) the angles which is subtended at the centre of the loop :

Options

A. `9.67^(@)`
B. `7.69^(@)`
C. `6.97^(@)`
D. none of these

Answer

To solve the problem step by step, we will follow the instructions provided in the video transcript and apply the relevant formulas.

### Step 1: Find the length of the circular wire
The length of the circular wire (S) can be calculated using the formula for the circumference of a circle:
\[ S = 2\pi r \]
where \( r \) is the radius of the circular wire. Given that the radius of the wire is 2.5 cm, we convert this to meters for consistency with the hoop's radius:
\[ S = 2\pi \times 0.025 \text{ m} \]
\[ S = 2\pi \times 0.025 = 0.15708 \text{ m} \]

### Step 2: Find the circumference of the hoop
The circumference (C) of the hoop can also be calculated using the same formula:
\[ C = 2\pi R \]
where \( R \) is the radius of the hoop. Given that the radius of the hoop is 1.29 m:
\[ C = 2\pi \times 1.29 \text{ m} \]
\[ C = 2\pi \times 1.29 = 8.094 \text{ m} \]

### Step 3: Set up the proportion to find the angle
We know that the angle \( \theta \) subtended at the center of the hoop can be found using the formula:
\[ \frac{S}{C} = \frac{\theta}{360^\circ} \]
Substituting the values of S and C into the equation:
\[ \frac{0.15708}{8.094} = \frac{\theta}{360} \]

### Step 4: Solve for \( \theta \)
To find \( \theta \), we rearrange the equation:
\[ \theta = \frac{0.15708}{8.094} \times 360 \]
Calculating the right side:
\[ \theta = 0.0194 \times 360 \]
\[ \theta \approx 6.97^\circ \]

### Final Answer
The angle subtended at the center of the hoop is approximately \( 6.97^\circ \).

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